Zeroes of Sine and Cosine

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Theorem

$(1): \quad \forall n \in \Z: x = \left({n + \dfrac 1 2}\right) \pi \implies \cos x = 0$
$(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$


Proof

From Sine and Cosine are Periodic on Reals: Corollary:


$\cos x$ is:

strictly positive on the interval $\displaystyle \left({-\frac \pi 2 \,.\,.\, \frac \pi 2}\right)$

and:

strictly negative on the interval $\displaystyle \left({\frac \pi 2 \,.\,.\, \frac {3 \pi} 2}\right)$


$\sin x$ is:

strictly positive on the interval $\left({0 \,.\,.\, \pi}\right)$

and:

strictly negative on the interval $\left({\pi \,.\,.\, 2 \pi}\right)$


The result follows directly from Sine and Cosine are Periodic on Reals.

$\blacksquare$